Now, that we have completely diagonalized the operator \({L}_{\perp }\)  in () up to error terms, we can find an almost approximate right inverse for it. This is done by almost inverting the operator \({L}_{\infty }\)  in (REF ). The result is stated below and its proof follows word by word [1]} or [2]}. Consequently, we omit the proof and only mention that the estimates mainly follows from (REF )-(REF )-()-(REF )-(REF ) and Lemma REF -(ii).
